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Earth geopotential effect on orbit propagation

I'm trying to take into account the Earth geopotential effect on the orbit.

  1. Are the terms (J2, etc.) change with time?
  2. Is it worth to take into account J0 and J1?
  3. In JGM-3 N=M=70 (about 5000 coefficients). I found till degree 8 here. Where to find others?
  4. I'm solving a differential equation with J2. How to add other terms?

    enter image description here

The shown equation gives 1 km error in comparison with GMAT, where degree 2 and order 0 is chosen for JGM-3 model. Probably, it takes into account J1 and J0?